Measuring and estimating

Every instrument has a least count: the smallest reading it can show. An ordinary 15 cm scale reads to 1 mm; an ordinary watch with a second hand, to 1 s. The smaller the least count, the more precise the instrument.

Two words that sound alike mean different things:

• Accuracy: how close a measurement is to the true value. It depends a lot on the person measuring.
• Precision: how finely the quantity is measured, which is set by the least count. 12.3 cm, 12.30 cm and 12.300 cm are the same number in mathematics, but each was read with a finer instrument than the one before.

What if a length is smaller than the least count? Measure it indirectly:

• Wind a thin wire or thread n times closely round a pencil, measure the whole width, and divide by n.
• Look at it through a microscope: the real size is the image size ÷ the magnification. Careful: if the image has 400 times the area, the linear magnification is only √400 = 20, because area magnification is the square of linear magnification.

'Large' and 'small' only mean something compared with a standard of the same kind: a large ant is still a tiny animal.

Readings always scatter a little. If five students measure the same table and get 2.34, 2.35, 2.31, 2.32 and 2.38 m, that is normal. If a sixth says 8.43 m, that is not an error but a mistake, and it is thrown out. Taking many readings and using the mean gives a more reliable value, because the small random errors above and below cancel out.